This could be described as the telescoped version of a word, i.e., shortened by omitting one or more syllables. Thus, Ruthven becomes "Rivven." But, perhaps it would be good to repeat the entire stanza
I once was as meek as a new-born lamb,
I'm now Sir Murgatroyd -- ha! ha!
With greater precision
(Without the elision),
Sir Ruthven Murgatroyd -- ha! ha!
In only this one place the name Ruthven is pronounced as spelled rather than as "Rivven." Reference to "Sir Murgatroyd" is a grammatical error because to be correct the "Sir" should be prefixed to a nobleman's Christian (i.e., first) name, not his family name. Thus "With greater precision" means "more correctly." In short, he might have said, "To speak more correctly, I'll apply the 'Sir' to my first name, which in this one instance I'll pronounce exactly the way it's spelled."
With greater precision -- without the elision